where \(\lambda \)  is the parameter of the model with a mass scale. This class of effective potentials works in particle physics and describes the interaction with other fields. The number \(n\)  is usually a positive integer e.g. \(n=2\)  corresponds to the well-known potential \(V=\frac{1}{2}M^{2}\varphi ^{2}\)  where \(M\)  is the mass of the inflaton and in the presence of the non-minimal coupling term (NMC) \(\xi \varphi ^{2}R\) , we deal with an effective mass \(m_{eff}=\sqrt{M^{2}+\xi R}\) . The case \(n=4\)  is related to \(V=\lambda \varphi ^{4}\)  and describes the chaotic inflation, where \(\lambda \)  is a self-interacting constant [1]}, [2]}. The cases \(n>6\)  seems excluded by the observations. Furthermore, one can find some fractional powers in the literature e.g \(n = 2/3\)  and \(n = 4/3\)  that could arise in axion monodromy inflation [3]}, [4]}, [5]}, [6]}, [7]}. Here, we consider the cases \(n=1,2,2/3,4/3\) .
